On the strength and independence number of powers of paths and cycles

A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set {1, 2, ... , n} to the vertices of G. The strength str (G) of G is defined by str (G) = min {strf (G) |f is a numbering of G }, where strf (G) = max {f (u) + f (v) |uv is an element of E (G) }. Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.